39 Americium-241 is an artificial isotope used in smoke de-
tectors. It undergoes alpha decay, with a half-life of 432 years. As
discussed in example 17 on page 907, alpha decay can be understood
as a tunneling process, and although the barrier is not rectangular
in shape, the equation for the tunneling probability on page 907
can still be used as a rough guide to our thinking. For americium-
241, the tunneling probability is about 1× 10 −^29. Suppose that
this nucleus were to decay by emitting a tritium (helium-3) nucleus
instead of an alpha particle (helium-4). Estimate the relevant tun-
neling probability, assuming that the total energyEremains the
same. This higher probability is contrary to the empirical observa-
tion that this nucleus is not observed to decay by tritium emission
with any significant probability, and in general tritium emission is
almost unknown in nature; this is mainly because the tritium nu-
cleus is far less stable than the helium-4 nucleus, and the difference
in binding energy reduces the energy available for the decay.
40 As far as we know, the mass of the photon is zero. However,
it’s not possible to prove by experiments that anything is zero; all
we can do is put an upper limit on the number. As of 2008, the
best experimental upper limit on the mass of the photon is about
1 × 10 −^52 kg. Suppose that the photon’s mass really isn’t zero, and
that the value is at the top of the range that is consistent with
the present experimental evidence. In this case, thecoccurring in
relativity would no longer be interpreted as the speed of light. As
with material particles, the speedvof a photon would depend on
its energy, and could never be as great asc. Estimate the relative
size (c−v)/cof the discrepancy in speed, in the case of a photon of
visible light.
.Answer, p. 1065
41 Hydrogen is the only element whose energy levels can be
expressed exactly in an equation. Calculate the ratioλE/λF of the
wavelengths of the transitions labeled E and F in problem 25 on
p. 945. Express your answer as an exact fraction, not a decimal
approximation. In an experiment in which atomic wavelengths are
being measured, this ratio provides a natural, stringent check on the
precision of the results.√42 Give a numerical comparison of the number of photons
per second emitted by a hundred-watt FM radio transmitter and
a hundred-watt lightbulb.√948 Chapter 13 Quantum Physics